Шексіз есептеуге келтірілетін функциялар шегі

The limit of a uniformly computable sequence of functions
In computability theory, a function is called limit computable if it is the limit of a uniformly computable sequence of functions. The terms computable in the limit, limit recursive and recursively approximable are also used. One can think of limit computable functions as those admitting an eventually correct computable guessing procedure at their true value. A set is limit computable just when its characteristic function is limit computable. If the sequence is uniformly computable relative to D, then the function is limit computable in D.

The limit lemma states that a set of natural numbers is limit computable if and only if the set is computable from (the Turing jump of the empty set). The relativized limit lemma states that a set is limit computable in if and only if it is computable from Moreover, the limit lemma (and its relativization) hold uniformly. Thus one can go from an index for the function to an index for relative to One can also go from an index for relative to to an index for some that has limit .

Iteration of limit computability can be used to climb up the arithmetical hierarchy. Namely, an ary function is iff it can be written in the form for some ary recursive function \(g\), under the assumption that all limits exist.

A real number x is computable in the limit if there is a computable sequence of rational numbers (or, which is equivalent, computable real numbers) which converges to x. In contrast, a real number is computable if and only if there is a sequence of rational numbers which converges to it and which has a computable modulus of convergence. When a real number is viewed as a sequence of bits, the following equivalent definition holds. An infinite sequence of binary digits is computable in the limit if and only if there is a total computable function taking values in the set such that for each i the limit exists and equals Thus for each i, as t increases the value of eventually becomes constant and equals As with the case of computable real numbers, it is not possible to effectively move between the two representations of limit computable reals.

The real whose binary expansion encodes the halting problem is computable in the limit but not computable. The real whose binary expansion encodes the truth set of first order arithmetic is not computable in the limit. Chaitin's constant.